Preconditioning in H(div) and Applications

Transcription

1 1 Preconditioning in H(div) and Applications Douglas N. Arnold 1, Ricard S. Falk 2 and Ragnar Winter 3 4 Abstract. Summarizing te work of [AFW97], we sow ow to construct preconditioners using domain decomposition and multigrid tecniques for te system of linear algebraic equations wic arises from te finite element discretization of boundary value problems associated to te differential operator I grad div. Tese preconditioners are sown to be spectrally equivalent to te inverse of te operator and tus may be used to precondition iterative metods so tat any given error reduction may be acieved in a finite number of iterations independent of te mes discretization. We describe applications of tese results to te efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems. 1.1 Introduction Tis paper summarizes te work of [AFW97], in wic we consider te solution of te system of linear algebraic equations wic arises from te finite element discretization of boundary value problems in two space dimensions for te differential operator I grad div. Te natural setting for te weak formulation of suc problems is te space: H(div) = { u L 2 (Ω) div u L 2 (Ω) }. Let (, ) denote te L 2 (Ω) inner product of bot scalar and vector-valued functions and J(u, v) := (u, v) + (div u, div v) 1 Department of Matematics, Penn State, University Park, PA dnamat.psu.edu 2 Dept. of Matematics, Rutgers University, New Brunswick, NJ falk@mat.rutgers.edu 3 Department of Informatics, University of Oslo, N-0316 Oslo, Norway. ragnar@ifi.uio.no 4 Te first autor was supported by NSF grants DMS and DMS and by te Institute for Matematics and its Applications. Te second autor was supported by NSF grant DMS Te tird autor was supported by Te Norwegian Researc Council under grants /431 and STP DD9 Proceedings Editor Petter Bjørstad, Magne Espedal and David Keyes c 1996 Jon Wiley & Sons Ltd.

2 2 D. N. ARNOLD, R. S. FALK, AND R. WINTHER denote te innerproduct on H(div). If f L 2 (Ω), te weak formulation is to find u H(div) suc tat for all v H(div), J(u, v) = (f, v). Tis corresponds to te boundary value problem (I grad div)u = f in Ω, div u = 0 on Ω. Note tat if u is a gradient, ten (I grad div)u = u+u, wile if u is a curl, ten (I grad div)u = u. A simple situation in wic te operator I grad div arises occurs in te computation of u = grad p, were p is te solution of te Diriclet problem p + p = g in Ω, p = 0 on Ω. Ten u H(div) satisfies J(u, v) = (g, div v) for all v H(div). Given a finite element subspace V of H(div), te natural finite element approximation sceme is: Find u V suc tat J(u, v ) = (f, v ) for all v V. We sall consider te case wen V consists of te Raviart Tomas space of index k 0, i.e., functions wic on eac triangle are of te form v(x, y) = p(x, y) + (x, y)q(x, y), p P k P k, q P k, (were P k denotes te polynomials of degree k) and for wic v n is continuous from triangle to triangle. Te goal is to find an efficient procedure for solving te discrete linear system corresponding to tis discretization, wic we write as J u = f. Denoting te eigenvalues of J by σ(j ), since te spectral condition number κ(j ) := max σ(j ) min σ(j ) of te operator J is O( 2 ), we will clearly need to precondition any standard iterative sceme if we want te number of iterations needed to acieve a given accuracy to be independent of. 1.2 Preconditioning in te abstract Let X L 2 be a finite dimensional normed vectorspace. We identify X and X as sets, but put te dual norm on te latter (dual wit respect to te L 2 inner product). Let A : X X be an L 2 -symmetric linear isomorpism. We suppose tat X is endowed wit an appropriate (energy) norm, i.e., we suppose tat A L(X,X ), A 1 L(X,X ) = O(1).

3 PRECONDITIONING IN H(DIV) AND APPLICATIONS 3 Given f X, we wis to solve A x = f by applying a standard iterative metod suc as CG or MINRES to te equation B A x = B f, were B : X X is an L 2 -symmetric, positive definite preconditioner. Our goal is to define B so tat te action of B is easily computable and κ(b A ) is bounded uniformly wit respect to. Since max σ(b A ) B A L(X,X ) A L(X,X ) B L(X,X ) and 1 min σ(b A ) (B A ) 1 L(X,X ) A 1 L(X,X ) B 1 L(X,X ) B is an effective preconditioner if B L(X,X ), B 1 L(X,X ) = O(1). In oter words, B is an effective preconditioner if it as te same mapping properties as A 1. Note tat te energy norm, and not te detailed structure of A, determine tese properties. Tus to solve te problem J u = f, we need to construct an efficiently computable operator K : V V for wic K L(V,V ), K 1 L(V,V ) = O(1). We will sow ow tis can be done using domain decomposition and multigrid tecniques. 1.3 Applications We are interested in te operator I grad div not for its own sake, but for its appearance in several important problems. Besides te example mentioned in te introduction, we will restrict our attention to two problems: te least squares formulation and te mixed formulation of second order scalar elliptic problems. Oter examples are discussed in [AFW97]. We first discuss te least squares variational principle. Consider te elliptic boundary value problem div(a grad p) = g in Ω, p = 0 on Ω, were te coefficient matrix A is assumed measurable, bounded, symmetric, and uniformly positive definite on Ω. Introducing u = A grad p leads to te first order system u A grad p = 0 in Ω, div u = g in Ω, p = 0 on Ω. Te least squares variational principle caracterizes te solution (u, p) as te minimizer of te functional v A grad q 2 + div v g 2

4 4 D. N. ARNOLD, R. S. FALK, AND R. WINTHER over te space H(div) H 1, were denotes te L 2 (Ω) norm and H 1 denotes te subspace of functions in H 1 (Ω) wic vanis on te boundary of Ω. Equivalently, we ave te weak formulation were B(u, p; v, q) = (g, div v) for all (v, q) H(div) H 1, B(u, p; v, q) = (u A grad p, v A grad q) + (div u, div v). To discretize te least squares formulation, we let X = V W be a finitedimensional subspace of H(div) H 1. Ten x := (u, p ) is te minimizer over X of v A grad q 2 + div v g 2, or in weak form, B(u, p ; v, q) = (g, div v) for all (v, q) X. Defining A : X X by (A x, y) = B(x, y) and f X by (f, (v, q)) = (g, div v), we may rewrite our problem as A x = f. Te key to te convergence teory for te least squares metod is te following teorem (cf. Pelivanov, Carey, Lazarov [PCL94] and Cai, Lazarov, Manteuffel, and McCormick [CLMM94]). Teorem 1.1 Te bilinear form B is an inner product on H(div) H 1 equivalent to te usual one. A direct consequence of te teorem is tat A : X X is symmetric, positive definite and satisfies A L(X,X ), A 1 L(X,X ) = O(1). Tus we need a preconditioner wit te opposite mapping properties. Since X = V W, we can coose a block diagonal preconditioner ( ) K 0 B =, 0 M were K is a good preconditioner in H(div), i.e., it maps like J 1 : V V, and M is a good preconditioner in H 1, i.e., it maps like 1 : W W. Hence we conclude tat a good preconditioner for te discrete least squares system is obtained using an H(div) preconditioner for te vector variable and a standard H 1 preconditioner for te scalar variable. We next consider a mixed variational formulation of tis boundary value problem. Te mixed variational principle caracterizes (u, p) as a saddle point of over H(div) L 2, or, in weak form, 1 2 (A 1 v, v) + (q, div v) (g, q), (A 1 u, v) + (p, div v) = 0 for all v H(div),

5 PRECONDITIONING IN H(DIV) AND APPLICATIONS 5 (div u, q) = (g, q) for all q L 2. Coosing X = V S H(div) L 2, we can define a discrete solution x = (u, p ) V S by restricting eiter te variational or weak formulation. Tis may be written A x = f, wit A : X X L 2 -symmetric but indefinite, since A as te form A = ( a b b t 0 ). Te convergence of tis metod depends on te coice of V ypoteses for te convergence analysis are te Brezzi conditions: and S. Te key (A 1 v, v) γ 1 v H(div) for all v V wit div v S, inf q S (q, div v) sup γ 2. v V q v H(div) Tese conditions are satisfied if V is te Raviart Tomas space of index k and S te space of (discontinuous) piecewise polynomials of degree k. Brezzi s teorem states tat if bot ypoteses are satisfied, ten A is an isomorpism and A 1 L(X,X ) may be bounded in terms of te γ i. We tus base our coice of B on te discrete version of te isomorpism ( ) A grad : H(div) L 2 H(div) L 2. div 0 We again use a simple block-diagonal preconditioner, wic tis time takes te form ( ) K 0 B =, 0 I were I is te identity on S and again K is a good preconditioner in H(div), i.e., it maps like J 1 : V V. We remark tat most oter work on preconditioning suc mixed metods uses te alternate isomorpism ( ) A grad : L 2 div 0 H 1 L 2 H 1, wic leads to a different (and less natural) coice of preconditioner. 1.4 An additive Scwarz preconditioner for J We let T H = {Ω n } N n=0, denote te coarse mes and T a refinement (te fine mes). We let {Ω n} N n=1 be an overlapping covering aligned wit te fine mes suc tat Ω n Ω n. We make te standard assumption of sufficient but bounded overlap. Let V n denote te Raviart Tomas space approximating H(div, Ω n) wit te boundary condition v n = 0 on Ω j \ Ω. Let V 0 denote te Raviart Tomas approximation to H(div, Ω) using te coarse mes.

6 6 D. N. ARNOLD, R. S. FALK, AND R. WINTHER Given f V, define u n V n by J(u n, v) = (f, v) for all v V n. Te additive Scwarz preconditioner is ten defined by K f := N n=0 u n. Our main result for tis domain decomposition preconditioner is te following teorem (cf. [AFW97] for te proof). Teorem 1.2 Tere exists a constant c independent of bot and H for wic κ(k J ) c. Following te teoretical framework of Dryja Widlund [DW90] or Xu [Xu92], a critical step of te proof is te following decomposition lemma. Lemma 1.1 For all v V, tere exist v n V n wit v = N n=0 v n and N v n 2 H(div) c v H(div). n=0 Te standard proof uses a partition of unity {θ n } N n=1 and takes v 0 V 0 a suitable approximation of v and v n = Π [θ n (v v 0 )] wit Π a suitable local projection into V. Te analysis leads to te following estimates. div v n c div[θ n (v v 0 )] c grad θ n L v v 0 + θ n L div(v v 0 ) ch 1 v v 0 + div(v v 0 ). In te standard elliptic case we bound te first term using v v 0 CH v 1. However it is not true tat v v 0 CH v H(div), so tis approac fails. We are able to get around tis problem by using a discrete Helmoltz decomposition, wic we now describe. Let V denote te Raviart Tomas space of index k, S te space of piecewise polynomials of degree k, and W te space of C 0 piecewise polynomials of degree k + 1. Ten we ave te following discrete Helmoltz decomposition. V = curl W grad S, were grad : S V is defined by (grad s, v) = (s, div v). Returning to te decomposition lemma, we write v = curl w + grad s and observe tat v 2 H(div) = curl w 2 + grad s 2 H(div). We ten decompose eac term separately. Since curl w H(div) w 1, we can use te standard decomposition lemma on w to write n n w = w j, w j 2 1 c w 2 1. j=0 j=0 Taking curls gives us te desired result on te curl w term. For v = grad s and v 0 = grad H s 0, were (s 0, v 0 ) is te mixed metod approximation to (s, v) in te space S 0 V 0, we can prove using standard results from te teory of mixed finite element approximations tat v v 0 CH v H(div),

7 PRECONDITIONING IN H(DIV) AND APPLICATIONS 7 and conclude te proof. Te key is tat altoug te above estimate does not old for all v V, it does old wen v = grad s. 1.5 V-cycle preconditioner We consider a nested sequence of meses, T 1, T 2,..., T N, and let V n be te Raviart- Tomas space of some fixed order subordinate to te mes T n. Tis gives a nested sequence of spaces V 1 V 2 V N = V and corresponding operators J n : V n V n. We also require smooters R n : V n V n wic we discuss below and te H(div)- projection operators P n : H(div) V n. Multigrid ten defines K n : V n V n recursively starting wit K 1 = J 1 1. We sall make use of te following multigrid convergence result. Teorem 1.3 Suppose tat for eac n = 1, 2,..., N te smooter R n is L 2 - symmetric and positive semi-definite and satisfies te conditions J([I R n J n ]v, v) 0 (R 1 n [I P n 1 ]v, [I P n 1 ]v) αj([i P n 1 ]v, [I P n 1 ]v). Ten tere exists a constant C independent of and N suc tat te eigenvalues of K J lie in te interval [1 δ, 1] were δ = C/(C + 2m), m denoting te number of smootings. For standard elliptic operators many smooters can be sown to satisfy te ypoteses, te simplest of wic is te scalar smooter. However, te proof for te scalar smooter and some oters fails in H(div) and te multigrid preconditioner constructed wit tese smooters is not effective. We sall consider an additive Scwarz smooter, defined in te following way. For eac vertex of te mes, consider te patc of elements containing tat vertex. Tese patces form an overlapping covering of Ω and so determine an additive Scwarz operator. We use tis operator as our smooter. Te verification of te first ypotesis is routine. Te standard proof of te second fails, but te difficulty can be surmounted by again using te discrete Helmoltz decomposition in a manner similar to tat used for te proof of domain decomposition. Te complete proof is given in [AFW97]. 1.6 Numerical Results First we made a numerical study of te condition number of J and te effect of preconditioning. In Table 1.1, te level m mes is a uniform triangulation of te unit square into 2 2m 1 triangles and as mes size = 1/2 m 1. Te space V is taken as te Raviart Tomas space of index 0 on tis mes. Te preconditioner K is te V-cycle multigrid preconditioner using one application of te standard additive Scwarz smooter wit te scaling factor taken to be 1/2. Te fift column of te table clearly displays te expected growt of te condition number of J as O( 2 ),

8 8 D. N. ARNOLD, R. S. FALK, AND R. WINTHER Table 1.1 Condition numbers for te operator J and for te preconditioned operator K J, and iterations counts to acieve an error reduction factor of level elements dim V κ(j ) κ(k J ) iterations / / / , / , /32 2,048 3,136 42, /64 8,192 12,416 8 and te sixt column te boundedness of te condition number of te preconditioned operator K J. As a second numerical study, we used te Raviart Tomas mixed metod to solve te factored Poisson equation u = grad p, div u = g in Ω, p = 0 on Ω, again on te unit square using te same sequence of meses as in te first example. We cose g = 2(x 2 +y 2 x y) so tat p = (x 2 x)(y 2 y). Te discrete solution (u, p ) belongs to te space V S, wit V te Raviart Tomas space described above and S te space of piecewise constant functions on te same mes. We solved te discrete equations bot wit a direct solver and by using te minimum residual metod preconditioned wit te block diagonal preconditioner aving as diagonal blocks K and te identity (as discussed previously). Full multigrid was used to initialize te minimum residual algoritm. Tat is, te computed solution at eac level was used as an initial guess at te next finer level, beginning wit te exact solution on te coarsest (two element) mes. In Table 1.2, we sow te condition number of te discrete operator A and of te preconditioned operator B A. Wile te former quantity grows linearly wit 1 (since tis is a first order system), te latter remains small. Table 1.2 Condition numbers for te indefinite operator A corresponding to te mixed system and for te preconditioned operator B A. level dim V dim S κ(a ) κ(b A ) / / / /

9 References [AFW97] Arnold D. N., Falk R. S., and Winter R. (1997) Preconditioning in H(div) and applications. Matematics of Computation to appear. [CLMM94] Cai Z., Lazarov R., Manteuffel T., and McCormick S. (1994) First-order system least squares for second-order partial differential equations: Part I. SIAM J. Numer. Anal. 31: [DW90] Dryja M. and Widlund O. B. (1990) Towards a unified teory of domain decomposition algoritms for elliptic problems. In Can T. F., Glowinski R., Périaux J., and Widlund O. B. (eds) Proc. Tird Int. Conf. on Domain Decomposition Mets., pages SIAM, Piladelpia. [PCL94] Pelivanov A. I., Carey G. F., and Lazarov R. D. (1994) Least-squares mixed finite elements for second-order elliptic problems. SIAM J. Numer. Anal. 31: [Xu92] Xu J. (1992) Iterative metods by space decomposition and subspace correction. SIAM Rev. 34: DD9 Proceedings Editor Petter Bjørstad, Magne Espedal and David Keyes c 1996 Jon Wiley & Sons Ltd.